In a small saucepan combine the water, butter, granulated sugar, and salt and bring the mixture to a rapid boil. Remove the pan from the heat and add the flour all at once, stirring vigorously. Cook the paste over low heat, beating briskly, until the ingredients are thoroughly combined and the dough cleanly leaves the sides of the pan and forms a ball. Remove the pan from the heat. Add the eggs, 1 at a time, beating well after each addition. By hand or with an electric mixer set at medium speed beat the paste until it is smooth and glossy. Stir in the vanilla.
In the deep fryer, heat 3 inches shortening 370 degrees F. Drop the dough by teaspoonfuls into the shortening, and fry the beignets in batches, turning them, until golden brown (about 3 minutes). With a slotted spoon, remove to paper towels to drain. Sprinkle the beignets with the confectioners' sugar and serve them hot.
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